薛定谔方程有限元超收敛研究

发布时间：2018-03-31 00:31

本文选题：薛定谔方程　切入点：超收敛　出处：《湘潭大学》2016年博士论文

【摘要】：薛定谔方程是量子力学最基本的方程,在非线性光学、等离子物理、电磁波理论、核物理、量子化学等领域中被广泛应用.薛定谔方程也是量子力学的一个基本假定,并不能从什么比它更根本的假定来证明它,它的正确性只能靠实践来检验,而且在实际中复杂的薛定谔方程不易求得精确解.因此,关于其数值解的研究越来越受到重视和关注.本文在均匀剖分的矩形网格或广义矩形网格上,主要研究了三类薛定谔方程的有限元超收敛结果.在第三章中,针对二维含时线性薛定谔方程,首先在空间上用双p次Lagrange矩形有限元得到半离散格式,分别利用有限元插值误差估计理论和椭圆投影算子进行误差分析,得到了半离散数值解与精确解的插值函数之间具有超收敛结果,并构造插值后处理算子得到了整体超收敛.然后在时间方向用Crank-Nicolson方法得到全离散格式,证明了全离散数值解与精确解的插值函数之间具有超收敛结果,通过数值算例验证了理论结果.在第四章中,针对二维不含时非线性薛定谔方程,用双线性矩形有限元将原问题进行离散,利用椭圆投影算子对有限元数值解进行了误差分析,得到了超收敛结果,同样运用插值后处理技术得到了整体超收敛,并用数值算例验证了理论结果的正确性.在第五章中,针对二维含时非线性薛定谔方程,首先在空间方向用双线性矩形有限元得到半离散格式,利用椭圆投影算子证明了半离散数值解与精确解的插值函数之间具有超收敛结果,通过插值后处理技术得到了整体超收敛.然后在时间上用向后Euler方法和C rank-Nicolson方法得到两种全离散格式,证明了这两种全离散格式的数值解与精确解在L2范数下的最优阶误差估计,最后用数值算例验证了理论结果.
[Abstract]:Schrodinger equation is the most basic equation in quantum mechanics. It is widely used in nonlinear optics, plasma physics, electromagnetic wave theory, nuclear physics, quantum chemistry and so on.Schrodinger equation is also a basic assumption of quantum mechanics. It can not be proved by what is more fundamental than it. Its correctness can only be verified by practice, and the complex Schrodinger equation is not easy to obtain exact solution in practice.Therefore, more and more attention has been paid to the numerical solution.In this paper, the finite element superconvergence results of three classes of Schrodinger equations are studied on uniformly divided rectangular or generalized rectangular meshes.In the third chapter, for the two-dimensional time-dependent linear Schrodinger equation, the semi-discrete scheme is obtained by using the double p-degree Lagrange rectangular finite element method in space, and the error analysis is carried out by using the interpolation error estimation theory of finite element method and the elliptic projection operator, respectively.The superconvergence results between the interpolation functions of semi-discrete numerical solutions and exact solutions are obtained, and the global superconvergence is obtained by constructing interpolation post-processing operators.Then the full discrete scheme is obtained by using the Crank-Nicolson method in the time direction. It is proved that there is superconvergence between the interpolation function of the full discrete numerical solution and the exact solution, and the theoretical results are verified by a numerical example.In chapter 4, for the two-dimensional time-free nonlinear Schrodinger equation, the original problem is discretized by the bilinear rectangular finite element method. The error analysis of the finite element numerical solution is carried out by using elliptical projection operator, and the superconvergence result is obtained.The global superconvergence is also obtained by the interpolation post-processing technique, and the correctness of the theoretical results is verified by a numerical example.In the fifth chapter, for the two-dimensional time-dependent nonlinear Schrodinger equation, the semi-discrete scheme is obtained by bilinear rectangular finite element method in the space direction.By using elliptic projection operator, it is proved that the interpolation function of semi-discrete numerical solution and exact solution has superconvergence results, and the global superconvergence is obtained by interpolation post-processing technique.Then, two kinds of fully discrete schemes are obtained by backward Euler method and C rank-Nicolson method in time, and the optimal order error estimates of the numerical solutions and exact solutions of these two schemes under L 2 norm are proved. Finally, the theoretical results are verified by numerical examples.
【学位授予单位】：湘潭大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82;O413.1

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